Cambridge IGCSE Mathematics (with coursework) 0581 — 2011 Oct/Nov Paper 2 · Variant 1

This document consists of 12 printed pages. IB11 11_0581_21/FP © UCLES 2011 [Turn over *8821023100* UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MATHEMATICS 0581/21 Paper 2 (Extended) October/November 2011 1 hour 30 minutes Candidates answer on the Question Paper. Additional Materials: Electronic calculator Geometrical instruments Mathematical tables (optional) Tracing paper (optional) READ THESE INSTRUCTIONS FIRST Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. DO NOT WRITE IN ANY BARCODES. Answer all questions. If working is needed for any question it must be shown below that question. Electronic calculators should be used. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For π, use either your calculator value or 3.142. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total of the marks for this paper is 70. www.XtremePapers.com

2 © UCLES 2011 0581/21/O/N/11 For Examiner's Use 1 Use your calculator to find 1.5 3.1 5.75 45 + × . Answer [2] 2 Work out 2(3 × 108 – 4 × 106), giving your answer in standard form. Answer [2] 3 Write the following in order of size, largest first. sin 158° cos 158° cos 38° sin 38° Answer K K K [2] 4 Write down all the working to show that . 6 1 3 3 2 5 3 3 2 5 3 = × + Answer [3]

3 © UCLES 2011 0581/21/O/N/11 [Turn over For Examiner's Use 5 A circle has a radius of 50 cm. (a) Calculate the area of the circle in cm2. Answer(a) cm2 [2] (b) Write your answer to part (a) in m2. Answer(b) m2 [1] 6 NOT TO SCALE The front of a house is in the shape of a hexagon with two right angles. The other four angles are all the same size. Calculate the size of one of these angles. Answer [3]

4 © UCLES 2011 0581/21/O/N/11 For Examiner's Use 7 t ° y° z° 50° O B A T NOT TO SCALE TA is a tangent at A to the circle, centre O. Angle OAB = 50°. Find the value of (a) y, Answer(a) y = [1] (b) z, Answer(b) z = [1] (c) t. Answer(c) t = [1] 8 Seismic shock waves travel at speed v through rock of density d. v varies inversely as the square root of d. v = 3 when d = 2.25. Find v when d = 2.56. Answer v = [3]

5 © UCLES 2011 0581/21/O/N/11 [Turn over For Examiner's Use 9 O A The point A lies on the circle centre O, radius 5 cm. (a) Using a straight edge and compasses only, construct the perpendicular bisector of the line OA. [2] (b) The perpendicular bisector meets the circle at the points C and D. Measure and write down the size of the angle AOD. Answer(b) Angle AOD = [1]

6 © UCLES 2011 0581/21/O/N/11 For Examiner's Use 10 In a flu epidemic 45% of people have a sore throat. If a person has a sore throat the probability of not having flu is 0.4. If a person does not have a sore throat the probability of having flu is 0.2. Sore throat No sore throat Flu No flu No flu Flu 0.45 0.4 0.2 Calculate the probability that a person chosen at random has flu. Answer [4] 11 Work out. (a) 2 3 4 1 2       Answer(a)         [2] (b) 1 3 4 1 2 −       Answer(b)         [2]

7 © UCLES 2011 0581/21/O/N/11 [Turn over For Examiner's Use 12 50 40 30 20 10 0 10 20 30 40 Mathematics test mark English test mark 50 60 70 80 The scatter diagram shows the marks obtained in a Mathematics test and the marks obtained in an English test by 15 students. (a) Describe the correlation. Answer(a) [1] (b) The mean for the Mathematics test is 47.3 . The mean for the English test is 30.3 . Plot the mean point (47.3, 30.3) on the scatter diagram above. [1] (c) (i) Draw the line of best fit on the diagram above. [1] (ii) One student missed the English test. She received 45 marks in the Mathematics test. Use your line to estimate the mark she might have gained in the English test. Answer(c)(ii) [1]

8 © UCLES 2011 0581/21/O/N/11 For Examiner's Use 13 C A B D O a b A and B have position vectors a and b relative to the origin O. C is the midpoint of AB and B is the midpoint of AD. Find, in terms of a and b, in their simplest form (a) the position vector of C, Answer(a) [2] (b) the vector . Answer(b) [2] 14 T = 2π g l (a) Find T when g = 9.8 and ℓ = 2. Answer(a) T = [2] (b) Make g the subject of the formula. Answer(b) g = [3]

9 © UCLES 2011 0581/21/O/N/11 [Turn over For Examiner's Use 15 A container ship travelled at 14 km/h for 8 hours and then slowed down to 9 km/h over a period of 30 minutes. It travelled at this speed for another 4 hours and then slowed to a stop over 30 minutes. The speed-time graph shows this voyage. 16 14 12 10 8 6 4 2 Speed (km / h) 0 1 2 3 4 5 6 Time (hours) 7 8 9 10 11 12 13 (a) Calculate the total distance travelled by the ship. Answer(a) km [4] (b) Calculate the average speed of the ship for the whole voyage. Answer(b) km/h [1]

10 © UCLES 2011 0581/21/O/N/11 For Examiner's Use 16 y x O A (1,3) B (3,0) C (9,7) NOT TO SCALE The co-ordinates of A, B and C are shown on the diagram, which is not to scale. (a) Find the length of the line AB. Answer(a) AB = [3] (b) Find the equation of the line AC. Answer(b) [3]

11 © UCLES 2011 0581/21/O/N/11 [Turn over For Examiner's Use 17 f(x) = 4 1 + x (x ≠ O4) g(x) = x2 – 3x h(x) = x3 + 1 (a) Work out fg(1). Answer(a) [2] (b) Find h O1(x). Answer(b) h O1(x) = [2] (c) Solve the equation g(x) = O2. Answer(c) x = or x = [3] Question 18 is printed on the next page.

12 Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. © UCLES 2011 0581/21/O/N/11 For Examiner's Use 18 The first four terms of a sequence are T1 = 12 T2 = 12 + 22 T3 = 12 + 22 + 32 T4 = 12 + 22 + 32 + 42 . (a) The nth term is given by Tn = 6 1 n(n + 1)(2n + 1). Work out the value of T23. Answer(a) T23 = [2] (b) A new sequence is formed as follows. U1 = T2 – T1 U2 = T3 – T2 U3 = T4 – T3 ……. (i) Find the values of U1 and U2. Answer(b)(i) U1 = and U2 = [2] (ii) Write down a formula for the nth term, Un . Answer(b)(ii) Un = [1] (c) The first four terms of another sequence are V1 = 22 V2 = 22 + 42 V3 = 22 + 42 + 62 V4 = 22 + 42 + 62 + 82. By comparing this sequence with the one in part (a), find a formula for the nth term, Vn . Answer(c) Vn = [2]

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the October/November 2011 question paper for the guidance of teachers 0581 MATHEMATICS 0581/21 Paper 2 (Extended), maximum raw mark 70 This mark scheme is published as an aid to teachers and candidates, to indicate the requirements of the examination. It shows the basis on which Examiners were instructed to award marks. It does not indicate the details of the discussions that took place at an Examiners’ meeting before marking began, which would have considered the acceptability of alternative answers. Mark schemes must be read in conjunction with the question papers and the report on the examination. • Cambridge will not enter into discussions or correspondence in connection with these mark schemes. Cambridge is publishing the mark schemes for the October/November 2011 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses. www.XtremePapers.com

Page 2 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – October/November 2011 0581 21 © University of Cambridge International Examinations 2011 Abbreviations cao correct answer only cso correct solution only dep dependent ft follow through after error isw ignore subsequent working oe or equivalent SC Special Case www without wrong working Qu. Answers Mark Part Marks 1 7.5(0) cao 2 M1 for 6.4 75 . 258 2 5.92 × 108 2 M1 figs 592 on answer line or M1 296 × 106 oe in working 3 cos38 sin38 sin158 cos158 2 M1 correct decimals seen 0.3(74..) –0.9(271..) 0.7(88..) 0.6(15..) 4 Answer given 3 M1 15 19 M115 6 or × 6 15 seen E1 6 1 3 6 19 = = 5 (a) 7853 to 7855 or 7850 or 7860 www 2 M1 for π × 502 (b) 0.7853 to 0.7855 or 0.785 or 0.786 1ft Their (a) ÷ 10 000 evaluated 6 135 cao 3 M1 for 720 or (6 – 2) × 180 oe seen in working and M1 for equation 180 + 4x = their 720 or M1 for (360 − 180) ÷ 4 (= 45) oe seen in working and M1 dep for 180 − their 45 7 (a) (y =) 80 1 (b) (z =) 40 1 (c) (t =) 10 1ft Follow through 90 − their y or 50 − their z 8 2.81(25) 3 M1 d k V = or M1 ( ) d k V = A1 k = 4.5 A1 k = 20.25 9 (a) Correct perpendicular bisector with arcs 2 B1 correct line B1 correct construction arcs (b) 60° 1 10 0.38 or 50 19 4 B1 0.8, 0.6 or 0.55 then M1 0.45 × their 0.6 M1 0.2 × their 0.55 or M2 1 – (0.45 × 0.4 + 0.55 × their 0.8)

Page 3 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – October/November 2011 0581 21 © University of Cambridge International Examinations 2011 11 (a)       13 20 5 8 2 B1 two or three entries correct (b)       − − 1 2 1 2 1 2 1 oe 2 B1       d b c a 2 1 B1( )       − − 2 4 1 3 k 12 (a) Negative 1 Ignore embellishments (b) Correct point 1 (c) (i) Accurate ruled line 1 (ii) English mark 1ft Follow through their (c)(i) 13 (a) 2 1 a + 2 1 b oe 2 M1 unsimplified or any correct route e.g a + 2 1 (b – a) or OA + AC (b) –1 2 1 a + 1 2 1 b oe 2 M1 unsimplified or any correct route e.g. CD = 1 2 1 AB or b – a + 2 1 (b – a) 14 (a) 2.84 2 M1 correct substitution of g and l seen (b) 2 2 π 4 T l oe 3 M1 each correct move but third move marked on answer line 15 (a) 156 4 M1 intention to find area under graph B2 completely correct area statement or B1 two areas found correctly (or one trapezium area) (b) 12 1ft Their (a)/13 16 (a) 3.61 3 M1 (3 – 1)2 + (0 – 3)2 oe M1 2 2 3 2 + (b) y = 2 1 x + 2 2 1 oe 3 B2 y = 2 1 x + k or y = kx + 2 2 1 or B1 kx + 2 2 1 or 2 1 x + k If 0 scored B1 m = 2 1 B1 c = 2 2 1 clearly identified in working

Page 4 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – October/November 2011 0581 21 © University of Cambridge International Examinations 2011 17 (a) 2 1 2 B1 f(–2) seen (b) 3√(x – 1) or 3 1 − x 2 M1 x –1 = y3 or 3√(y – 1) (c) 1 2 3 M2 (x – 1)(x – 2) = 0 or M1 (x + a)(x + b) = 0 where ab = 2 or a + b = –3 If 0 scored give M1 for x2 – 3x + 2 = 0 18 (a) 4324 cao 2 M1 6 1 × 23 × 24 × 47 or better (b) (i) 4, 9 2 B1 either correct (ii) (n + 1)2 or n2 + 2n + 1 1 (c) 3 2 n(n+ 1)(2n + 1) oe 2 M1 recognising Vn = 4Tn